Why are capacitors used when obtaining dual power supply from a single power supply and how to know the values?

What happens if we only use 2 resistors in the figure. Why do we use capacitor if we have a DC source already? And whats the easy way to calculate capacitor capacitance for given R and Vin?

At the end of the day, this circuit forms a low pass filter. The corner frequency is: $f_o = \dfrac{1}{2\pi RC}$. The capacitors are there to help stabilize the virtual ground. Practically $1-10\mu F$ with $R=10k\Omega$ is a good place to start. If your virtual ground is noisy relative to the system ground, look at the spectral content, and adjust R and C values to minimize it.

If you don't use the capacitors and rely on the resistors, any variation in ground current will create a variation in the centre voltage of the two resistors and this will shift +V and -V up and down. They'll move together because Vin is a constant supply. It's basically ohms law - you draw more current through 1 resistor and it will drop more voltage and shift the offset from half-way.

If the undulations in the ground current are high enough frequency, capacitors can be chosen that minimize the shifting on +V and -V. This is because capacitors oppose the change in voltage and will supply the current (temporarily) without altering their terminal voltage very much.

Q = CV and therefore $\dfrac{dQ}{dt}$ = $C\dfrac{dv}{dt}$

Change in charge ($\dfrac{dQ}{dt}$) = current so $I=C\dfrac{dv}{dt}$

If C is larger the change in voltage ($\dfrac{dv}{dt}$) is smaller.

A better solution (favoured in op-amp circuits) is to dedicate an op-amp as a unity gain follower to keep the "centre" voltage constant - useful for up to +/- 20mA draw from the op-amp output but some op-amps will be able to supply +/-100mA or more whilst keeping the centre-point stable in voltage.

Footnote - you can get by with one capacitor if Vin is a good supply because it acts like a large capacitance across +V and -V but, if unsure about Vin then use 2 caps.

• Whether or not one needs a voltage follower depends upon what is done with the virtual ground. There are some situations where the virtual ground feeds nothing except some op amp non-inverting inputs, in which case a voltage follower may not help much. It may also be worth noting that if supplies are noisy the caps may be helpful for ensuring that the virtual ground stays at the mid-rail point. Many circuits may have parasitic capacitances to the positive or negative rails; using decently-matched moderate-sized caps may swamp the effects of imbalances in the parasitic caps. – supercat Nov 15 '13 at 17:35
• What? You absolutely do need the capacitors, even if Vin is an ideal voltage source. The impedance between the virtual ground and either rail is defined by the resistors, and will be non-zero regardless of how good Vin is. This is precisely not what we want of a voltage source: adding the capacitors lower the virtual ground impedance for AC. – Phil Frost Nov 15 '13 at 20:38
• @PhilFrost I think the key word is "both" - you need at least one but not both - should i make that clearer do ya think Phil? – Andy aka Nov 15 '13 at 21:42
• @Andyaka yes, I see what you mean now. A clarification would be good. – Phil Frost Nov 15 '13 at 22:25